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chore(MathSpec.md): small fix in constant values, try fixing formatting to github display it fine

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Ricardo Guilherme Schmidt
2024-10-20 12:09:00 -03:00
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docs/MathSpec.md
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@@ -23,7 +23,7 @@
Minimum lockup period, equivalent to 90 days.
- $T_{MAX_{YEARS}} = \pu{4 \mathrm{years}}$
Maximum years of lockup period.
- $T_{MAX} = T_{MAX_{YEARS}} \times T_{YEAR} = \pu{126230400 \mathrm{s}}$
- $T_{MAX} = T_{MAX_{YEARS}} \times T_{YEAR} = \pu{126227700 \mathrm{s}}$
Maximum lockup period in seconds.
## Variables
### System and User Parameters
@@ -42,16 +42,25 @@
Seconds $a_{bal}$ remains locked, expressed as:
$$
\begin{aligned}
&t_{lock, \Delta} = max(t_{lock,end},t_{now}) - t_{now} \\ \small\text{ where: }\normalsize\quad & t_{lock, \Delta} = 0\;\text{ or }\;T_{MIN} \le t_{lock, \Delta} \le T_{MAX}\end{aligned}$$
&t_{lock, \Delta} = max(t_{lock,end},t_{now}) - t_{now} \\ \small\text{ where: }\normalsize\quad & t_{lock, \Delta} = 0\;\text{ or }\;T_{MIN} \le t_{lock, \Delta} \le T_{MAX}\end{aligned}
$$
### State Related
- $a_{bal}\rightarrow$ = Amount of Balance
Amount of tokens in balance, where $a_{bal} \ge A_{MIN}$.
- $t_{lock,end}\rightarrow$ = Time Lock End
Timestamp marking the end of the lock period, it state can be defined as: $$t_{lock,end} = \max(t_{now}, t_{lock,end}) + t_{lock}$$ The value of $t_{lock,end}$ can be updated only within the functions:
Timestamp marking the end of the lock period, it state can be defined as:
$$
t_{lock,end} = \max(t_{now}, t_{lock,end}) + t_{lock}
$$
The value of $t_{lock,end}$ can be updated only within the functions:
- $\mathcal{f}^{stake}(\tiny\mathbb{Account}\normalsize, \Delta a, \Delta t_{lock})$;
- $\mathcal{f}^{lock}(\tiny\mathbb{Account}\normalsize, \Delta t_{lock})$;
- $t_{last}\rightarrow$ Time of Accrual
Timestamp of the last accrued time, it state can be defined as:$$t_{last} = t_{now}$$The value of $t_{last}$ can is updated by all functions that change state:
Timestamp of the last accrued time, it state can be defined as:
$$
t_{last} = t_{now}
$$
The value of $t_{last}$ is updated by all functions that change state:
- $f^{accrue}(\tiny\mathbb{Account}\normalsize, a_{bal},\Delta t)$,
- $\mathcal{f}^{stake}(\tiny\mathbb{Account}\normalsize, \Delta a, \Delta t_{lock})$;
- $\mathcal{f}^{lock}(\tiny\mathbb{Account}\normalsize, \Delta t_{lock})$;
@@ -65,7 +74,8 @@
- $\mathcal{f}^{unstake}(\tiny\mathbb{Account}\normalsize, \Delta a)$.
It's state can be expressed as the following state changes:
1. Increase in balance and lock: $$
1. Increase in balance and lock:
$$
\begin{aligned}
mp_\mathcal{M} &= mp_\mathcal{M} + mp_\mathcal{A}(\Delta a, M_{MAX} \times T_{YEAR}) \\
&\quad + mp_\mathcal{B}(\Delta a, t_{lock,\Delta} + t_{lock}) \\
@@ -73,15 +83,22 @@
&\quad + mp_\mathcal{I}(\Delta a)
\end{aligned}
$$
2. Increase in balance only: $$
\begin{aligned} mp_\mathcal{M} &= mp_\mathcal{M} + mp_\mathcal{A}(\Delta a, M_{MAX} \times T_{YEAR}) \\ &\quad + mp_\mathcal{B}(\Delta a, t_{lock,\Delta}) \\ &\quad + mp_\mathcal{I}(\Delta a) \end{aligned}
2. Increase in balance only:
$$
3. Increase in lock only: $$
\begin{aligned} mp_\mathcal{M} &= mp_\mathcal{M} + mp_\mathcal{B}(a_{bal}, t_{lock}) \end{aligned}
\begin{aligned}
mp_\mathcal{M} &= mp_\mathcal{M} + mp_\mathcal{A}(\Delta a, M_{MAX} \times T_{YEAR}) \\
&\quad + mp_\mathcal{B}(\Delta a, t_{lock,\Delta}) \\
&\quad + mp_\mathcal{I}(\Delta a)
\end{aligned}
$$
3. Increase in lock only:
$$
mp_\mathcal{M} &= mp_\mathcal{M} + mp_\mathcal{B}(a_{bal}, t_{lock})
$$
4. Decrease in balance:
$$
mp_\mathcal{M} &= mp_\mathcal{M} - mp_\mathcal{R}(mp_\mathcal{M}, a_{bal}, \Delta a)
$$
4. Decrease in balance: $$
\begin{aligned} mp_\mathcal{M} &= mp_\mathcal{M} - mp_\mathcal{R}(mp_\mathcal{M}, a_{bal}, \Delta a) \end{aligned}
$$
- $mp_{\Sigma}\rightarrow$ Total Multiplier Points
Altered by all functions that change state:
- $\mathcal{f}^{stake}(\tiny\mathbb{Account}\normalsize, \Delta a, \Delta t_{lock})$;
@@ -134,25 +151,33 @@
\end{aligned} \right\}
$$
$\mathbb{Account} = \{ a_{bal}, t_{lock,end}, t_{last}, mp_\Sigma, mp_\mathcal{M} \}$$\Delta t_{lock} = \max(t_{lock,end}, t_{now}) - t_{now} \quad \text{where} \quad 0 \leq \Delta t_{lock} \leq T_{MAX} \quad \text{or} \quad \Delta t_{lock} = 0$
$
---
## Pure Mathematical Functions
>[!info] This function definitions represent direct mathematical input -> output methods, which don't change state.
### Definition: $\mathcal{f}{mp_\mathcal{I}}(\Delta a) \longrightarrow$ Initial Multiplier Points
Calculates the initial multiplier points (**MPs**) based on the balance change $\Delta a$. The result is equal to the amount of balance added.
$$
\boxed{\begin{equation}\mathcal{f}{mp_\mathcal{I}}(\Delta a) = \Delta a\end{equation}}
\boxed{
\begin{equation}
\mathcal{f}{mp_\mathcal{I}}(\Delta a) = \Delta a
\end{equation}
}
$$
Where
- **$\Delta a$**: Represents the change in balance.
---
### Definition: $\mathcal{f}{mp_\mathcal{A}}(a_{bal}, \Delta t) \longrightarrow$ Accrue Multiplier Points
Calculates the accrued multiplier points (**MPs**) over a time period **$\Delta t$**, based on the account balance **$a_{bal}$** and the annual percentage yield $MP_{APY}$.
$$
\boxed{
\begin{equation}
\mathcal{f}mp_\mathcal{A}(a_{bal}, \Delta t) = \dfrac{a_{bal} \times \Delta t \times MP_{APY}}{100 \times T_{YEAR}}
\end{equation}
\begin{equation}
\mathcal{f}mp_\mathcal{A}(a_{bal}, \Delta t) = \dfrac{a_{bal} \times \Delta t \times MP_{APY}}{100 \times T_{YEAR}}
\end{equation}
}
$$
Where
@@ -160,13 +185,19 @@ Where
- **$\Delta t$**: The time difference or the duration over which the multiplier points are accrued, expressed in the same time units as the year (typically days or months).
- **$T_{YEAR}$**: A constant representing the duration of a full year, used to normalize the time difference **$\Delta t$**.
- **$MP_{APY}$**: The Annual Percentage Yield (APY) expressed as a percentage, which determines how much the balance grows over a year.
---
### Definition: $\mathcal{f}{mp_\mathcal{B}}(\Delta a, t_{lock}) \longrightarrow$ Bonus Multiplier Points
Calculates the bonus multiplier points (**MPs**) earned when a balance **$\Delta a$** is locked for a specified duration **$t_{lock}$**. It is equivalent to the accrued multiplier points function $\mathcal{f}mp_\mathcal{A}(\Delta a, t_{lock})$ but specifically applied in the context of a locked balance.
$$
\begin{aligned}
&\;\mathcal{f}mp_\mathcal{B}(\Delta a, t_{lock}) = \mathcal{f}mp_\mathcal{A}(\Delta a, t_{lock}) \\
&\boxed{\begin{equation}\mathcal{f}mp_\mathcal{B}(\Delta a, t_{lock}) = \dfrac{\Delta a \times t_{lock} \times MP_{APY}}{100 \times T_{YEAR}}\end{equation}}
&\boxed{
\begin{equation}
\mathcal{f}mp_\mathcal{B}(\Delta a, t_{lock}) = \dfrac{\Delta a \times t_{lock} \times MP_{APY}}{100 \times T_{YEAR}}
\end{equation}
}
\end{aligned}
$$
Where:
@@ -178,86 +209,214 @@ Where:
### Definition: $\mathcal{f}{mp_\mathcal{R}}(mp, a_{bal}, \Delta a) \longrightarrow$ Reduce Multiplier Points
Calculates the reduction in multiplier points (**MPs**) when a portion of the balance **$\Delta a$** is removed from the total balance **$a_{bal}$**. The reduction is proportional to the ratio of the removed balance to the total balance, applied to the current multiplier points **$mp$**.
$$
\boxed{\begin{equation}\mathcal{f}{mp_\mathcal{R}}(mp, a_{bal}, \Delta a) = \dfrac{mp \times \Delta a}{ a_{bal}}\end{equation}}
\boxed{
\begin{equation}
\mathcal{f}{mp_\mathcal{R}}(mp, a_{bal}, \Delta a) = \dfrac{mp \times \Delta a}{ a_{bal}}
\end{equation}
}
$$
Where:
- **$mp$**: Represents the current multiplier points.
- **$a_{bal}$**: The total account balance before the removal of **$\Delta a$**.
- **$\Delta a$**: The amount of balance being removed or deducted.
---
## State Functions
These function definitions represent methods that modify the state of both **$\mathbb{System}$** and **$\mathbb{Account}$**. They perform various pure mathematical operations to implement the specified state changes, affecting either the system as a whole and the individual account states.
### Definition: $\mathcal{f}_{stake}(\tiny\mathbb{Account}\normalsize,\Delta a, t_{lock}) \longrightarrow$ Stake Amount With Lock
_Purpose:_ Allows a user to stake an amount $\Delta a$ with an optional lock duration $t_{lock}$.
#### Steps
1. Calculate the New Remaining Lock Period ($\Delta t_{lock}$): $$\Delta t_{lock} = max(\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end}, t_{now}) + t_{lock} - t_{now}$$
1. Calculate the New Remaining Lock Period ($\Delta t_{lock}$):
$$
\Delta t_{lock} = max(\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end}, t_{now}) + t_{lock} - t_{now}
$$
2. Verify Constraints:
- Ensure new balance ($a_{bal}$ + $\Delta a$) meets the minimum amount ($A_{MIN}$): $$\tiny\mathbb{Account}\normalsize _\cdot a_{bal} + \Delta a > A_{MIN}$$
- Ensure the New Remaining Lock Period ($\Delta t_{lock}$) is within allowed limits ($T_{MIN}$ and $T_{MAX}$):$$\Delta t_{lock} = 0\;\lor\;T_{MIN} \le \Delta t_{lock} \le T_{MAX}$$
- Ensure new balance ($a_{bal}$ + $\Delta a$) meets the minimum amount ($A_{MIN}$):
$$
\tiny\mathbb{Account}\normalsize _\cdot a_{bal} + \Delta a > A_{MIN}
$$
- Ensure the New Remaining Lock Period ($\Delta t_{lock}$) is within allowed limits ($T_{MIN}$ and $T_{MAX}$):
$$
\Delta t_{lock} = 0\;\lor\;T_{MIN} \le \Delta t_{lock} \le T_{MAX}
$$
3. Accrue Existing Multiplier Points (MPs):
- Call the $\mathcal{f}_{accrue}(\tiny\mathbb{Account}\normalsize)$ function to update MPs and last accrual time.
4. Calculate Increased Bonus MPs ($\Delta \hat{mp}_\mathcal{B}$) for the Increased Balance ($\Delta a$) and Increased Lock Period ($t_{lock}$):
- For the new amount ($\Delta a$) with the New Remaining Lock Period ($\Delta t_{lock}$): $$\Delta \hat{mp}_\mathcal{B} = \mathcal{f}mp_\mathcal{B}(\Delta a, \Delta t_{lock})$$
- For extending the lock ($t_{lock}$) on the existing balance ($\tiny\mathbb{Account}\normalsize _\cdot a_{bal}$): $$\Delta \hat{mp}_\mathcal{B} = \Delta \hat{mp}_\mathcal{B} + \mathcal{f}mp_\mathcal{B}(\tiny\mathbb{Account}\normalsize _\cdot a_{bal}, t_{lock})$$
5. Calculate Increased Maximum MPs ($\Delta mp_\mathcal{M}$): $$\Delta mp_\mathcal{M} = \mathcal{f}mp_\mathcal{I}(\Delta a) + \Delta \hat{mp}_\mathcal{B} + \mathcal{f}mp_\mathcal{A}(\Delta a, M_{MAX} \times T_{YEAR})$$
6. Calculate Increased Total MPs ($\Delta mp_\Sigma$): $$\Delta mp_\Sigma = \mathcal{f}mp_\mathcal{I}(\Delta a) + \Delta \hat{mp}_\mathcal{B}$$
- For the new amount ($\Delta a$) with the New Remaining Lock Period ($\Delta t_{lock}$):
$$
\Delta \hat{mp}_\mathcal{B} = \mathcal{f}mp_\mathcal{B}(\Delta a, \Delta t_{lock})
$$
- For extending the lock ($t_{lock}$) on the existing balance ($\tiny\mathbb{Account}\normalsize _\cdot a_{bal}$):
$$
\Delta \hat{mp}_\mathcal{B} = \Delta \hat{mp}_\mathcal{B} + \mathcal{f}mp_\mathcal{B}(\tiny\mathbb{Account}\normalsize _\cdot a_{bal}, t_{lock})
$$
5. Calculate Increased Maximum MPs ($\Delta mp_\mathcal{M}$):
$$
\Delta mp_\mathcal{M} = \mathcal{f}mp_\mathcal{I}(\Delta a) + \Delta \hat{mp}_\mathcal{B} + \mathcal{f}mp_\mathcal{A}(\Delta a, M_{MAX} \times T_{YEAR})
$$
6. Calculate Increased Total MPs ($\Delta mp_\Sigma$):
$$
\Delta mp_\Sigma = \mathcal{f}mp_\mathcal{I}(\Delta a) + \Delta \hat{mp}_\mathcal{B}
$$
7. Update account state
- Update account maximum MPs: $$\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{Account}\normalsize\cdot mp_\mathcal{M} + \Delta mp_\mathcal{M}$$
- Update account total MPs: $$\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma + \Delta mp_\Sigma$$
- Update account balance: $$\tiny\mathbb{Account}\normalsize _\cdot a_{bal} = \tiny\mathbb{Account}\normalsize _\cdot a_{bal} + \Delta a$$
- Update account lock end time: $$\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end} = max(\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end}, t_{now}) + t_{lock}$$
- Update account maximum MPs:
$$
\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{Account}\normalsize\cdot mp_\mathcal{M} + \Delta mp_\mathcal{M}
$$
- Update account total MPs:
$$
\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma + \Delta mp_\Sigma
$$
- Update account balance:
$$
\tiny\mathbb{Account}\normalsize _\cdot a_{bal} = \tiny\mathbb{Account}\normalsize _\cdot a_{bal} + \Delta a
$$
- Update account lock end time:
$$
\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end} = max(\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end}, t_{now}) + t_{lock}
$$
8. Update system state
- Update system maximum MPs: $$\tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} + \Delta mp_\mathcal{M}$$
- Update system total MPs: $$\tiny\mathbb{System}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{System}\normalsize _\cdot mp_\Sigma + \Delta mp_\Sigma$$
- Update system total staked amount: $$\tiny\mathbb{System}\normalsize _\cdot a_{bal} = \tiny\mathbb{System}\normalsize _\cdot a_{bal} + \Delta a$$
- Update system maximum MPs:
$$
\tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} + \Delta mp_\mathcal{M}
$$
- Update system total MPs:
$$
\tiny\mathbb{System}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{System}\normalsize _\cdot mp_\Sigma + \Delta mp_\Sigma
$$
- Update system total staked amount:
$$
\tiny\mathbb{System}\normalsize _\cdot a_{bal} = \tiny\mathbb{System}\normalsize _\cdot a_{bal} + \Delta a
$$
---
### Definition: $\mathcal{f}_{lock}(\tiny\mathbb{Account}\normalsize, t_{lock}) \longrightarrow$ Increase Lock
> [!info] Equivalent to $\mathcal{f}_{stake}(\tiny\mathbb{Account}\normalsize,0, t_{lock})$
_Purpose:_ Allows a user to lock the $\tiny\mathbb{Account}\normalsize _\cdot a_{bal}$ with a lock duration $t_{lock}$.
#### Steps
1. Calculate the New Remaining Lock Period ($\Delta t_{lock}$): $$\Delta t_{lock} = max(\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end}, t_{now}) + t_{lock} - t_{now}$$
1. Calculate the New Remaining Lock Period ($\Delta t_{lock}$):
$$
\Delta t_{lock} = max(\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end}, t_{now}) + t_{lock} - t_{now}
$$
2. Verify Constraints:
- Ensure the New Remaining Lock Period ($\Delta t_{lock}$) is within allowed limits:$$\Delta t_{lock} = 0\;\lor\;T_{MIN} \le \Delta t_{lock} \le T_{MAX}$$
- Ensure the New Remaining Lock Period ($\Delta t_{lock}$) is within allowed limits:
$$
\Delta t_{lock} = 0\;\lor\;T_{MIN} \le \Delta t_{lock} \le T_{MAX}
$$
3. Accrue Existing Multiplier Points (MPs):
- Call the $\mathcal{f}_{accrue}(\tiny\mathbb{Account}\normalsize)$ function to update MPs and last accrual time.
4. Calculate Bonus MPs for the Increased Lock Period: $$\Delta \hat{mp}_\mathcal{B} = mp_\mathcal{B}(\tiny\mathbb{Account}\normalsize _\cdot a_{bal}, t_{lock})$$
4. Calculate Bonus MPs for the Increased Lock Period:
$$
\Delta \hat{mp}_\mathcal{B} = mp_\mathcal{B}(\tiny\mathbb{Account}\normalsize _\cdot a_{bal}, t_{lock})
$$
5. Update account state:
- Update maximum MPs: $$\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} + \Delta \hat{mp}_\mathcal{B}$$
- Update total MPs: $$\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma + \Delta \hat{mp}_\mathcal{B}$$
- Update lock end time: $$\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end} = max(\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end}, t_{now}) + t_{lock}$$
- Update maximum MPs:
$$
\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} + \Delta \hat{mp}_\mathcal{B}
$$
- Update total MPs:
$$
\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma + \Delta \hat{mp}_\mathcal{B}
$$
- Update lock end time:
$$
\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end} = max(\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end}, t_{now}) + t_{lock}
$$
6. Update system state:
- Update system maximum MPs: $$\tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} + \Delta mp_\mathcal{B}$$
- Update system total MPs: $$\tiny\mathbb{System}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{System}\normalsize _\cdot mp_\Sigma + \Delta mp_\mathcal{B}$$
- Update system maximum MPs:
$$
\tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} + \Delta mp_\mathcal{B}
$$
- Update system total MPs:
$$
\tiny\mathbb{System}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{System}\normalsize _\cdot mp_\Sigma + \Delta mp_\mathcal{B}
$$
---
### Definition: $\mathcal{f}_{unstake}(\tiny\mathbb{Account}\normalsize, \Delta a) \longrightarrow$ Unstake Amount Unlocked
1. Verify constraints:
- Ensure the account is not locked: $$\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end} < t_{now}$$
- Ensure that account have enough balance: $$\tiny\mathbb{Account}\normalsize _\cdot a_{bal} > \Delta a$$
- Ensure that new balance ($\tiny\mathbb{Account}\normalsize _\cdot a_{bal} - \Delta a$) will be zero or more than minimum allowed: $$\tiny\mathbb{Account}\normalsize _\cdot a_{bal} - \Delta a = 0\;\lor\; \tiny\mathbb{Account}\normalsize _\cdot a_{bal} - \Delta a > A_{MIN}$$
- Ensure the account is not locked:
$$
\tiny\mathbb{Account}\normalsize _\cdot t_{lock,end} < t_{now}
$$
- Ensure that account have enough balance:
$$
\tiny\mathbb{Account}\normalsize _\cdot a_{bal} > \Delta a
$$
- Ensure that new balance ($\tiny\mathbb{Account}\normalsize _\cdot a_{bal} - \Delta a$) will be zero or more than minimum allowed:
$$
\tiny\mathbb{Account}\normalsize _\cdot a_{bal} - \Delta a = 0\;\lor\; \tiny\mathbb{Account}\normalsize _\cdot a_{bal} - \Delta a > A_{MIN}
$$
2. Accrue Existing Multiplier Points (MPs):
- Call the $\mathcal{f}_{accrue}(\tiny\mathbb{Account}\normalsize)$ function to update MPs and last accrual time.
3. Calculate reduced amounts:
- For maximum MPs: $$\Delta mp_\mathcal{M} =\mathcal{f}mp_\mathcal{R}(\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M}, \tiny\mathbb{Account}\normalsize _\cdot a_{bal}, \Delta a)$$
- For total MPs: $$\Delta mp_\Sigma = \mathcal{f}mp_\mathcal{R}(\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma, \tiny\mathbb{Account}\normalsize _\cdot a_{bal}, \Delta a)$$
- For maximum MPs:
$$
\Delta mp_\mathcal{M} =\mathcal{f}mp_\mathcal{R}(\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M}, \tiny\mathbb{Account}\normalsize _\cdot a_{bal}, \Delta a)
$$
- For total MPs:
$$
\Delta mp_\Sigma = \mathcal{f}mp_\mathcal{R}(\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma, \tiny\mathbb{Account}\normalsize _\cdot a_{bal}, \Delta a)
$$
4. Update account state:
- Update maximum MPs: $$\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} - \Delta mp_\mathcal{M}$$
- Update total MPs:$$\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma - \Delta mp_\Sigma$$
- Update balance: $$\tiny\mathbb{Account}\normalsize _\cdot a_{bal} = \tiny\mathbb{Account}\normalsize _\cdot a_{bal} - \Delta a$$
- Update maximum MPs:
$$
\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} - \Delta mp_\mathcal{M}
$$
- Update total MPs:
$$
\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma - \Delta mp_\Sigma
$$
- Update balance:
$$
\tiny\mathbb{Account}\normalsize _\cdot a_{bal} = \tiny\mathbb{Account}\normalsize _\cdot a_{bal} - \Delta a
$$
5. Update system state:
- Update system maximum MPs: $$\tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} - \Delta mp_\mathcal{M}$$
- Update system total MPs:$$\tiny\mathbb{System}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{System}\normalsize _\cdot mp_\Sigma - \Delta mp_\Sigma$$
- Update system total staked amount:$$\tiny\mathbb{System}\normalsize _\cdot a_{bal} = \tiny\mathbb{System}\normalsize _\cdot a_{bal} - \Delta a$$
- Update system maximum MPs:
$$
\tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} = \tiny\mathbb{System}\normalsize _\cdot mp_\mathcal{M} - \Delta mp_\mathcal{M}
$$
- Update system total MPs:
$$
\tiny\mathbb{System}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{System}\normalsize _\cdot mp_\Sigma - \Delta mp_\Sigma
$$
- Update system total staked amount:
$$
\tiny\mathbb{System}\normalsize _\cdot a_{bal} = \tiny\mathbb{System}\normalsize _\cdot a_{bal} - \Delta a
$$
---
### Definition: $\mathcal{f}_{accrue}(\tiny\mathbb{Account}\normalsize) \longrightarrow$ Accrue Multiplier Points
1. Calculate the time period since last accrual:$$\Delta t = t_{now} - \tiny\mathbb{Account}\normalsize _\cdot t_{last}$$
1. Calculate the time period since last accrual:
$$
\Delta t = t_{now} - \tiny\mathbb{Account}\normalsize _\cdot t_{last}
$$
2. Verify constraints:
- Ensure the accrual period is greater than the minimum rate period:$$\Delta t > T_{RATE}$$
4. Calculate accrued MP for the accrual period:$$\Delta \hat{mp}_\mathcal{A} = min(\mathcal{f}mp_\mathcal{A}(\tiny\mathbb{Account}\normalsize _\cdot a_{bal},\Delta t) ,\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} - \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma)$$
- Ensure the accrual period is greater than the minimum rate period:
$$
\Delta t > T_{RATE}
$$
4. Calculate accrued MP for the accrual period:
$$
\Delta \hat{mp}_\mathcal{A} = min(\mathcal{f}mp_\mathcal{A}(\tiny\mathbb{Account}\normalsize _\cdot a_{bal},\Delta t) ,\tiny\mathbb{Account}\normalsize _\cdot mp_\mathcal{M} - \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma)
$$
5. Update account state:
- Update total MPs: $$\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma + \Delta \hat{mp}_\mathcal{A}$$
- Update last accrual time: $$\tiny\mathbb{Account}\normalsize _\cdot t_{last} = \tiny\mathbb{Account}\normalsize _\cdot t_{now}$$
- Update total MPs:
$$
\tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{Account}\normalsize _\cdot mp_\Sigma + \Delta \hat{mp}_\mathcal{A}
$$
- Update last accrual time:
$$
\tiny\mathbb{Account}\normalsize _\cdot t_{last} = \tiny\mathbb{Account}\normalsize _\cdot t_{now}
$$
6. Update system state:
- Update system total MPs: $$\tiny\mathbb{System}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{System}\normalsize _\cdot mp_\Sigma + \Delta \hat{mp}_\mathcal{A}$$
- Update system total MPs:
$$
\tiny\mathbb{System}\normalsize _\cdot mp_\Sigma = \tiny\mathbb{System}\normalsize _\cdot mp_\Sigma + \Delta \hat{mp}_\mathcal{A}
$$
---